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We study the global dimension function$\operatorname{gldim}\colon\operatorname{Aut}\backslash\operatorname{Stab}\mathcal{D}/\mathbb{C}\to\mathbb{R}_{\ge0}$on a quotient of the space of Bridgeland stability conditions on a triangulatedcategory $\mathcal{D}$ as well as Toda's Gepner equatio$\Phi(\sigma)=s\cdot\sigma$ for some $\sigma\in\operatorname{Stab}\mathcal{D}$and $(\Phi,s)\in\operatorname{Aut}\mathcal{D}\times\mathbb{C}$.  For the bounded derived category $\mathcal{D}^b(\mathbf{k} Q)$ of a Dynkinquiver $Q$, we show that there is a unique minimal point $\sigma_G$ of$\operatorname{gldim}$ (up to the $\mathbb{C}$-action), with value $1-2/h$,which is the solution of the Gepner equation $\tau(\sigma)=(-2/h)\cdot\sigma$.Here $\tau$ is the Auslander-Reiten functor and $h$ is the Coxeter number. Thissolution $\sigma_G$ was constructed by Kajiura-Saito-Takahashi. We also showthat for an acyclic non-Dynkin quiver $Q$, the minimal value of$\operatorname{gldim}$ is $1$.  Our philosophy is that the infimum of $\operatorname{gldim}$ on$\operatorname{Stab}\mathcal{D}$ is the global dimension for the triangulatedcategory $\mathcal{D}$. We explain how this notion could shed light on thecontractibility conjecture of the space of stability conditions.

Global dimension function on stability conditions and Gepner equations